Lcomoment.Lk12 {lmomco} | R Documentation

## Compute a Single Sample L-comoment

### Description

Compute the k >= 1 order L-comoment (λ_{k[r:n]}) for a given pair of random variables. The order of the L-comoments is specified.

### Usage

    Lcomoment.Lk12(X1,X2,k=1)

### Arguments

`X1` | An vector of random variables (a sample of random variable 1).
`X2` | Another vector of random variables (a sample of random variable 2).
`k` | The order of the L-comoment to compute. The default is 1.

### Details

L-comoments of random variable X1 are computed from the concomitants of X2. That is, X2 is sorted in ascending order to create the order statistics of X2. X1 is in turn reshuffled to the order of X2 for form the concomitants of X2 (denoted as X^{(12)}). The concomitants inturn are used in a weighted summation and expectation calculation to compute the L-comoment of X1 with respect to X2. The inverse can also be done (`Lcomoment.Lk12(X2,X1,k=1)`) and is not necessarily equal to (`Lcomoment.Lk12(X1,X2,k=1)`). The notation of `Lk12` is to read “Lambda for kth order L-comoment”, where the `12` portion of the notation reflects that of Serfling and Xiao (2006). The weights for the computation are derived from calls by `Lcomoment.Lk12` to `Lcomoment.Wk`.

hat{λ}_{k[12]} = frac{1}{n}sum_{r=1}^{n} w^{(k)}_{r:n} x^{(12)}_{[r:n]}

The L-comoments of X2 are computed from the concomitants of X1 (X^{(21)}) are formed by sorting X1 in ascending order and in turn shuffling X2 by the order of X1. The sample concomitants are thus formed (x^{(12)}_{[r:n]}). By symmetry the L-comoment is

hat{λ}_{k[21]} = frac{1}{n}sum_{r=1}^{n} w^{(k)}_{r:n} x^{(21)}_{[r:n]}

### Value

A single L-comoment.

### Note

The function begins with a capital letter. This is intentionally done so that lower case namespace is preserved. L-comoments are new in the literature and experimental in this package. By using a capital letter now, then `lcomoment.Lk12` or similar remains an available name in future releases.

### Author(s)

W.H. Asquith

### Source

Serfling and Xiao (2006).

### References

Hosking, J.R.M. and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

Serfling, R., and Xiao, P., 2006, Multivariate L-moments, preprint.

### See Also

`Lcomoment.matrix`, `Lcomoment.Wk`

### Examples

    X1   <- rnorm(20)
    X2   <- rnorm(20)
    Lk12 <- Lcomoment.Lk12(X1,X2,k=1)

[Package lmomco version 0.92 Index]